1. A K-Main Routes Approach to Spatial Network Activity Summarization
Authors:
Dev Oliver
Shashi Shekhar
James M. Kang
Renee Bousselaire
Abdussalam Bannur

2. Outline Motivation Problem Statement Contributions Validation
Analytical
Experimental
Case Studies
Summary and Future Work

3. Motivation: Crime Analysis (application domain)
Street
Place
Neighborhood
Crime hotspot
Area of concentrated crime
**J. E. Eck et. al. Mapping Crime: Understanding Hot Spots. US National Inst. of Justice (
“Most clustering algorithms will show areas of concentration even when a line is the most appropriate dimension.” – National Institute of Justice**
Star Tribune, January 26, 2011

4. Examples of Linear Patterns
Linear patterns resulting from deforestation in Brazil
Linear patterns of crime in a major US city

5. Motivation: Environmental Criminology (scientific domain)
Spatial theories in Environmental Criminology
Routine Activity Theory1
Crime location related to criminal’s frequently visited areas
Crime Pattern Theory2
Based on spatial model
Nodes (e.g. home, work, entertainment),
Paths (e.g. routes between nodes),
Edges
Crime locations close to edges
Near criminal’s activity boundaries where residents may not recognize him/her
Source: Rossmo, Kim (2000). Geographic Profiling. Boca Raton, FL: CRC Press.
Network based summarization adds value to Environmental Criminology
Assist with large scale verification of real-world data matching theories
Opportunities to develop hypotheses for new theory formulation
1L.E. Cohen et al., Social change and crime rate trends: A routine activity approach, American sociological review, 1979.
2P. L. Brantingham et al., Environmental Criminology, Waveland Press, 1990.

6. Other Domains
Disaster Relief
Accident Analysis and Prevention

7. Each edge has a weight of 1
Key Concepts
Activity
Object of interest located at node or edge
Summary path
A path chosen by KMR to summarize activities
Activity coverage
Total number of activities of a path or set of paths
Active node
A node having n ≥ 1 activities or joined by an edge having n ≥ 1 activities e.g., A, B, C, D, E
Inactive node
A node having n = 0 activities and joined by edges all having n = 0 activities e.g., F
Active node ratio
Total # active nodes/Total # nodes
e.g., 5/6
Each edge has a weight of 1 7

8. Given P = the set of Shortest Paths
Problem Statement
Given P = the set of Shortest Paths
Given
A spatial network G = (N, E)
A set of activities, A and their locations (e.g. a node or edge)
A set of Paths, P
K (Number of routes)
Edge weights
Find
A cardinality k subset P′ of P, i.e., a subset P′⊆ P with |P′| = k
Objective
Maximize the activity coverage (AC) by P′
Constraints
1 ≤ k ≤ |P|.
k = 2
Edge Weights
are 1 8

9. Challenges Measures of interestingness Computational Complexity
Activity coverage, average distance, etc
Computational Complexity
Choose(N,2) paths, given N nodes
Exponential number of k subsets of paths 9

10. SNAS is NP-Complete (Proof Sketch)
Devising an NP-Completeness proof for decision problem Π [1]
Show that Π is in NP
Select a known NP-Complete Problem Π’
Construct a transformation f from Π’ to Π
Prove that f is a polynomial transformation
1M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness. WH freeman San Francisco, 1979.

11. Step 1: SNAS is in NP
Verify in polynomial time whether activity coverage of P’ ≥ B
SNAS Decision problem
Given
A spatial network G = (N, E)
A set of activities, A and their locations (e.g. a node or edge)
A set of Paths, P
K (Number of routes)
Edge weights
B (bound on number of activities)
Find
A cardinality k subset P′ of P, i.e., a subset P′⊆ P with |P′| = k
Objective
Activity coverage (AC) by P′ ≥ B
Constraints
1 ≤ k ≤ |P|.

12. Step 2: Select a known NP-Complete Problem
Maximum Coverage
Input
Sets s1, s2, …,sm (the sets may have some elements E = {e1, e2, …, en} in common)
A number k < min (m,n)
Output
k sets such that the maximum number of elements are covered, i.e. the union of the selected sets has maximal size.

13. Step 3: Construct a transformation f from Π to Π’ (1/3)
Known NP Complete Problem
Polynomial transformation
A new
Problem
Solution to
New Problem
Solution to NP-Complete Problem
Maximum Coverage Problem
SNAS

14. Step 3: Construct a transformation f from Π to Π’ (2/3)
Maximum coverage input to SNAS input
Impose a total order, TO, to m elements E = {e1, e2, …, en}
Convert each element in E into a node with one activity
Convert each set si to a path pi
Sort elements in si using TO
Add edge (eij, eij+1) ∀ j ∈ 1 …. |si|
Example
Maximum Coverage:
E = {e1, e2, e3, e5, e6}
K = 2
S1 = {e1, e2}
S2 = {e2, e3}
S3 = {e1, e2, e3}
S4 = {e5, e6}
KMR:
P = {(e1→e2), (e2→e3), (e1→e2→e3), (e5→e6)}
K = 2
Activity = {a1, a2, a3, a5, a6}
Activity node = {a1–e1, a2–e2, a3–e3, a5–e5, a6–e6}
Candidate Solutions: (e1→e2→e3), (e5→e6) e1 e2 e3 e5 e6

15. Step 3: Construct a transformation f from Π to Π’ (3/3)
SNAS output to maximum coverage output
For each K route, Ri, produced by SNAS, convert the activities on the route into elements and form a set Si
Example
Given the K Routes:
(e1→e2→e3), (e5→e6)
S1 = {e1, e2, e3}
S2 = {e5, e6}

16. Network Summarization by Grouping/Clustering
Related Work
Network Summarization by Grouping/Clustering
Zero or One routes
Multiple routes
Clumping (Okabe), e.g. NT-VCM (Shiode)
Max. Subgraph, e.g. path, tree (Buchin)
Our Work 16

17. Contributions K-Main Routes (KMR) algorithm
Finds a set of k routes to group activities
New design decisions added
Network Voronoi Activity assignment
Divide and Conquer Summary path recomputation
Spatial network activity summarization is shown to be NP-complete.
Analytically demonstrate correctness of design decisions and show cost analysis
Experimental evaluation of the various algorithms
Performance evaluated using synthetic and real world datasets
Case study comparing KMR with geometry based summarization 17

18. K-Main Routes (KMR) Algorithm
P = the set of Shortest Paths, K=2
K-Main Routes (KMR) Algorithm
K-Main Routes Algorithm
Select k paths as initial summary paths
Repeat
Form k clusters by assigning each activity to its closest summary path
Recompute summary path of each cluster
Until summary paths do not change
Design Decisions
Inactive node pruning
Network Voronoi Activity assignment
Divide and Conquer Summary path recomputation
The lower left graph shows 2 active nodes N7 and N8. With inactive node pruning, we would only need to calculate and store shortest paths between these 2 nodes, as opposed to calculating and storing shortest paths between all the nodes in the graph. 18

19. Design Decision: Inactive Node Pruning
Only consider paths between active nodes
Optimal solution will still be in this set
Given the set of shortest paths
20 shortest paths calculated and stored versus 30

20. Design Decision: Network Voronoi (NV) Activity Assignment
Goals
Form k clusters by assigning each activity to its closest summary path
Improve execution time of current assignment strategy
Example (execution trace) Next
K-Main Routes Algorithm
Select k shortest paths as initial summary paths
Repeat
Network Voronoi Activity Assignment
Recompute summary path of each cluster
Until summary paths do not change
K-Main Routes Algorithm
Select k shortest paths as initial summary paths
Repeat
Form k clusters by assigning each activity to its closest summary path
Recompute summary path of each cluster
Until summary paths do not change

21. Design Decision: Network Voronoi (NV) Activity AssignmentX
Open: X A E D H
Closed: X ∞ ∞ ∞ A B 3 4 C 7 8 D ∞
ACTIVITIES 1 9 2 10 1 2 3 4 5 6 7 8 9 10 A E D H AE DH ∞ E 5 6 F G H ∞
DISTANCE FROM ∞ ∞
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node

22. Design Decision: Network Voronoi (NV) Activity AssignmentX
Open: A E D H B
Closed: X A ∞ 1 ∞ A B 3 4 C 7 8 D
ACTIVITIES 1 9 2 10 1 2 3 4 5 6 7 8 9 10 A E D H AE DH E 5 6 F G H
DISTANCE FROM ∞ ∞
1 < 0?
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node

23. Design Decision: Network Voronoi (NV) Activity AssignmentX
Open: E D H B F
Closed: X A E 1 ∞ A B 3 4 C 7 8 D
ACTIVITIES 1 9 2 10 1 2 3 4 5 6 7 8 9 10 A E D H AE DH E 5 6 F G H
DISTANCE FROM ∞ 1 ∞
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node

24. Design Decision: Network Voronoi (NV) Activity AssignmentX
Open: D H B F C
Closed: X A E D 1 ∞ 1 A B 3 4 C 7 8 D
ACTIVITIES 1 9 2 10 1 2 3 4 5 6 7 8 9 10 A E D H AE DH E 5 6 F G H
DISTANCE FROM 1 ∞
1 < 0?
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node

25. Design Decision: Network Voronoi (NV) Activity AssignmentX
Open: H B F C G
Closed: X A E D H 1 1 A B 3 4 C 7 8 D
ACTIVITIES 1 9 2 10 1 2 3 4 5 6 7 8 9 10 A E D H AE DH E 5 6 F G H
DISTANCE FROM 1 ∞ 1
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node

26. Design Decision: Network Voronoi (NV) Activity AssignmentX
Open: B F C G
2 < 1?
Closed: X A E D H B 1 1 A B 3 4 C 7 8 D
ACTIVITIES 1 9 2 10 1 2 3 4 5 6 7 8 9 10 A E D H AE DH E 5 6 F G H
DISTANCE FROM 1 1 1 1
2 < 1?
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node 1 1

27. Design Decision: Network Voronoi (NV) Activity AssignmentX
Open: F C G
Closed: X A E D H B F 1 1 A B 3 4 C 7 8 D
ACTIVITIES 1 9 2 10 1 2 3 4 5 6 7 8 9 10 A E D H AE DH E 5 6 F G H
DISTANCE FROM 1 1 1 1
2 < 1?
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node 1 1

28. Design Decision: Network Voronoi (NV) Activity AssignmentX
Open: C G
Closed: X A E D H B F C 1 1 A B 3 4 C 7 8 D
ACTIVITIES 1 9 2 10 1 2 3 4 5 6 7 8 9 10 A E D H AE DH E 5 6 F G H
DISTANCE FROM 1 1 1 1
2 < 1?
Activity
Active Node
Inactive Node
Virtual Node
Summary Path
Edge weight = 1
Edge weight = 0
Closed Node 1 1 1 1 1 1

29. Design Decision: Network Voronoi (NV) Activity Assignment
Network Voronoi Activity Assignment algorithm
Input: Graph G = (N, E), a set of Activities A, a set of k Summary Paths, S
Output: A set of k clusters formed by assigning all ai ∈A to one si ∈S, where dist(ai, si) ≤ dist(ai, sj) and sj ∈S and sj ≠ si
1. Open ← all nodes ∈ S, Closed ← Ø
2. Tnodes ← all nodes ∈ S,
3. Tactivities ← activities on si ∈S
4. repeat
nc ← next node ∈ Open
remove nc from Open
Closed ← nc
X ← neighbors of nc
foreach xi ∈ X
if xi ∉ Tnodes and xi ∉ Closed
Tnodes ← xi
xi.prev ← nc,
xi.dist ← dist(xi, nc) + nc.dist
xi.sp ← nc.sp
else if xi ∈Tnodes
update xi if new dist < xi.dist
if xi ∉ Open
Open ← xi
Y ← activities on edge {nc, xi}
foreach yi ∈ Y
if yi ∉ Tactivities
Tactivities ← yi
yi.prev ← nc
yi.dist ← xi.dist
yi.sp ← xi.sp
else
update yi if new dist < yi.dist
until all active nodes ∈ Closed
return currentClusters

30. Design Decision: Divide and Conquer Summary PAth REcomputation
Goals
Recompute the summary path of each cluster
Improve execution time of current recomputation strategy
Example (execution trace) Next
K-Main Routes Algorithm
Select k shortest paths as initial summary paths
Repeat
Network Voronoi Activity Assignment
Divide and Conquer Summary path Recomputation Design Decision
Until summary paths do not change
K-Main Routes Algorithm
Select k shortest paths as initial summary paths
Repeat
Network Voronoi Activity Assignment
Recompute summary path of each cluster
Until summary paths do not change

31. Design Decision: Divide and Conquer Summary PAth REcomputation
Summary Path Recomputation Algorithm
Input: Graph G = (N, E), a set of Clusters, C
Output: A set of summary paths, S where si ∈S has max coverage for ci ∈ C and si ∈ ci
nextClusters ← Ø
foreach ci ∈ C
X ← active nodes of ci
maxP ← Ø
foreach xi ∈ X
foreach xj ∈ X
if (i ≠ j)
cP ← getSP(xi, xj)
if (maxP = Ø)
maxP ← cP
if (maxP.activities < cP.activities)
if (maxP ≠ ci.summaryPath
nextClusters ← maxP
else
nextClusters ← ci.summaryPath
return nextClusters A B C D E F G H 1 2 3 4 5 6 7 8 9 10
Activity
Active Node
Inactive Node
Summary Path
Edge weights are 1
Cluster

32. Validation Analytical Experimental Case studies
Cost analysis explaining computational savings
Experimental
Comparative analysis of KMR with various design decisions
Performed on real and synthetic data
Network voronoi activity assignment and divide and conquer summary path recomputation saves computational costs
Savings increase with number of nodes, routes, activities and active node ratio
Case studies
Qualitatively shows the usefulness of network based summarization on Crime data

33. Analytical Evaluation: Computational Analysis
KMR Execution Time = Number of Iterations × (Activity Assignment Cost + Summary Path Recomputation Cost)
TKMR = I × ([K × |A| × cost(ai,ci)] + [K × dc × |N|2])
TKMR_I = I × ([K × |A| × cost(ai,ci)] + [K × dc × (|N| × r)2])
TKMR_IAS = I × ([|E| + |N|×log |N|] [K × dc × (|N|/K × r)2])
I = Number of Iterations
K = Number of Clusters
A = Set of activities
cost(ai, ci) = Cost of calculating the distance between activity ai and cluster ci
dc = Cost of looking up a path
N = Set of Nodes
E = Set of Edges
r = active node ratio, 0 ≤ r ≤ 1

34. Experimental Evaluation
Variables
Synthetic Dataset
Real Dataset
#Nodes
#Routes
Measures
Java-based Simulator
Analysis
#Activities
Active Node Ratio
Candidates
KMR_I
KMR_IV
KMR_ID
KMR_IVD
Goal: Comparative analysis
Candidates: KMR with various design decisions
KMR_I – KMR with inactive node pruning
KMR_IV – KMR with inactive node pruning and Network voronoi activity assignment
KMR_ID – KMR with Divide and conquer summary path recomputation
KMR_IVD – KMR with all three design decisions
Measure: CPU time (Unix time command)
Platform: Mac Pro, 2 x Xeon Quad Core 2.26 GHz, 16 GB RAM
Variables: #Nodes, #Routes, #Activities, Active Node Ratio
Fixed Parameters: unit edge length
Datasets: Synthetic and Real (Haiti Earthquake) 34

35. Data Description and Characteristics
Synthetic Data
2010 Census TIGER/Line® Shapefiles used for road network
Activities randomly assigned to each edge
Real-world data: Haiti Data Set
Geospatial and Temporal Dataset describing recent events post-disaster
Dataset collected from Jan 12, 2010 to March 23, 2010
1,677 records
Characteristics
Attributes
Incident Title (e.g., “Food, Water, Tents needed…”)
Incident Date and Time
Location (City, port name)
Category (numeric category)
Latitude/Longitude
Sources
Crisis Map of Haiti -
OpenStreetMap - 35

36. Effect of Number of Nodes
Synthetic Data Set
Number of Activities = 1200
Active Node Ratio = 0.2
K = 2
Real Data Set
Number of Activities = 1206
Active Node Ratio =
K = 2
Trends:
Voronoi Activity assignment and divide and conquer summary path recomputation saves comp. costs
Savings increase with number of nodes

37. Effect of Number of Routes, K
Synthetic Data Set
Number of Nodes = 1000
Number of Activities = 1200
Active Node Ratio = 0.2
Real Data Set
Number of Nodes = 1000
Number of Activities = 202
Active Node Ratio = 0.219
Trends:
Voronoi Activity assignment and divide and conquer summary path recomputation saves comp. costs
Savings increase with number of routes

38. Effect of Number of Activities
Synthetic Data Set
Number of Nodes = 1000
Active Node Ratio = 0.2
K = 2
Trends:
Voronoi Activity assignment and divide and conquer summary path recomputation saves comp. costs
Savings increase with number of activities

39. Effect of Active Node Ratio
Synthetic Data Set
Number of Nodes = 1000
Number of Activities = 1200
K = 2
Trends:
Voronoi Activity assignment and divide and conquer summary path recomputation saves comp. costs
Savings increase with active node ratio

40. Case Study: Crime Analysis
Input (a set of crime incidents, k=5)
KMR Output
Crimestat K-Means (Euclidean distance)
Crimestat K-Means (Network distance)

41. Case Study: Crime Analysis
Input (a set of crime incidents, k=5)
KMR Output
Crimestat K-Means (Euclidean distance)
Crimestat K-Means (Network distance)

42. Case Study: Crime Analysis
Input (a set of crime incidents, k=5)
KMR Output
Crimestat K-Means (Euclidean distance)
Crimestat K-Means (Network distance)

43. Summary
Spatial network activity summarization was shown to be NP-complete.
K-Main Routes (KMR) algorithm and its design decisions described
Inactive node pruning
Network Voronoi Activity assignment
Divide and Conquer Summary path recomputation
Analytically demonstrated correctness of design decisions and cost analysis showed
Experimental evaluation
Performance evaluated using synthetic and real world datasets
Case study comparing KMR with geometry based summarization 43

44. Future Work Short Term Long Term Usefulness
When is it useful to domain professionals (crime analysts, emergency managers)?
For which use cases is the proposed solution appropriate?
For which geographies is the proposed solution appropriate?
Distance based objective function instead of coverage based
Overlapping paths
Long Term
Dynamically changing incidents
Edge lengths, e.g. activities on a small section of a long edge 44

45. Acknowledgements
Members of the Spatial Database and Spatial Data Mining Research Group, University of Minnesota, Twin-Cities.
This work was supported by grants from USARMY and USDOD.
Thank you for your time! Any questions or comments?